At this point, it is instructive to compare our model with another widely studied model, the double well Bose-Hubbard model (DW-BHM) [1]} which describes an atomic condensate confined in a double-well potential. The Hamiltonian of the DW-BHM can be written as:
\(\mathcal {H}^{\rm DW}=\mathcal {E}_l b^{\dagger }_l b_l +\mathcal {E}_r b^{\dagger }_r b_r +\frac{V_l}{2}b^{\dagger 2}_l b^2_l+\frac{V_r}{2}b^{\dagger 2}_r b^2_r -t(b^{\dagger }_l b_r +b^{\dagger }_r b_l)\,.\) 
